Journal of Mathematical Sciences

, Volume 218, Issue 3, pp 335–353 | Cite as

On Quasi-Nonuniform Estimates for Asymptotic Expansions in the Central Limit Theorem


Improved asymptotic expansions are constructed in terms of the Chebyshev–Hermite polynomials in the local form of the central limit theorem for sums of independent identically distributed random variables under the condition of absolute integrability of some positive powers of the the characteristic function of a summand. The influence of the requirements to the order of existing moments on the accuracy of approximation is discussed. Theoretical results are illustrated by the example of a particular shifted exponential distribution.


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  1. 1.
    H. Cramer, Mathematical Methods of Statistics, Princeton University Press, Princeton (1948).MATHGoogle Scholar
  2. 2.
    V. V. Petrov, Sums of Independent Random Variables, Nauka, Moscow (1972).MATHGoogle Scholar
  3. 3.
    V. V. Senatov, Central Limit Theorem: Accuracy of Approximation and Asymptotic Expansions, URSS, Moscow (2009).MATHGoogle Scholar
  4. 4.
    V. V. Senatov and V. N. Sobolev, “New forms of asymptotic expansions in the central limit theorem,” Theor. Probab. Appl., 57, No 1, 82–96 (2013).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    V. V. Senatov, “Some new asymptotical expansions in the central limit theorem,” Theor. Probab. Appl., to appear.Google Scholar
  6. 6.
    I. G. Shevtsova, Optimization of the structure of moments estimations of the accuracy of normal approximations for distributions of sums of independent random variables, Doctoral Thesis, Moscow State University (2013).Google Scholar
  7. 7.
    H. Prawitz, “Noch einige ungleichungen fur characteristische funktionen,” Scand. Actuar. J., No. 1, 49–73 (1991).Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

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